Difference between revisions of "Discontinuous derivative"
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+ | Consider the function (blue curve) | ||
+ | :<math> f: \mathbb{R} \to \mathbb{R}, x \mapsto | ||
+ | \begin{cases} | ||
+ | x^2\sin(1/x),& x\neq 0,\\ | ||
+ | 0,& x=0\,. | ||
+ | \end{cases} | ||
+ | </math> | ||
+ | <math>f</math> is a continous and differentiable function. | ||
+ | The derivative of <math>f</math> is the function (red curve) | ||
+ | :<math> | ||
+ | f': \mathbb{R} \to \mathbb{R}, x \mapsto | ||
+ | \begin{cases} | ||
+ | 2x\sin(1/x) - \cos(1/x), &x \neq 0,\\ | ||
+ | 0,& x=0\,. | ||
+ | \end{cases} | ||
+ | </math> | ||
+ | We observe that <math>f'(0) = 0</math>, but <math>\lim_{x\to0}f'(x)</math> does not exist. | ||
+ | |||
+ | Therefore, <math>f'</math> is an example of a derivative which is not continuous. | ||
+ | |||
<jsxgraph width="500" height="500"> | <jsxgraph width="500" height="500"> | ||
− | var board = JXG.JSXGraph.initBoard('jxgbox', {axis:true, boundingbox:[-2,2,2,-2]}); | + | var board = JXG.JSXGraph.initBoard('jxgbox', {axis:true, boundingbox:[-1/2,1/2,1/2,-1/2]}); |
+ | |||
+ | var g = board.create('functiongraph', ["2*x*sin(1/x) - cos(1/x)"], {strokeColor: 'red'}); | ||
+ | var f = board.create('functiongraph', ["x^2*sin(1/x)"], {strokeWidth:2}); | ||
+ | </jsxgraph> | ||
− | var | + | Here is another example: |
+ | :<math> | ||
+ | g: \mathbb{R} \to \mathbb{R}, x \mapsto | ||
+ | \begin{cases} | ||
+ | x^2(1-x)^2\sin(1/(\pi x(1-x)),& 0 < x < 1\\ | ||
+ | 0,& \mbox{otherwise} | ||
+ | \end{cases}\,. | ||
+ | </math> | ||
+ | |||
+ | <jsxgraph width="500" height="500" box="jxgbox2"> | ||
+ | var board = JXG.JSXGraph.initBoard('jxgbox2', {axis:true, boundingbox:[-1/2,0.08,1.5,-0.02]}); | ||
+ | |||
+ | var g_der = board.create('functiongraph', ["(0 < x && x < 1) ? ((sin((1 / ((PI * x) * (1 - x)))) * ((2 * (x * ((1 - x)^2))) - (2 * ((x^2) * (1 - x))))) - (((x^2) * ((1 - x)^2)) * (cos((1 / ((PI * x) * (1 - x)))) * (((PI * (1 - x)) - (PI * x)) / (((PI * x) * (1 - x))^2))))) : 0"], {strokeColor: 'red'}); | ||
+ | var g = board.create('functiongraph', ["(0 < x && x < 1) ? x^2*(1-x)^2*sin(1/(PI* x*(1-x))) : 0"], {strokeWidth:2}); | ||
</jsxgraph> | </jsxgraph> | ||
+ | |||
===The underlying JavaScript code=== | ===The underlying JavaScript code=== | ||
+ | First example: | ||
<source lang="javascript"> | <source lang="javascript"> | ||
+ | var board = JXG.JSXGraph.initBoard('jxgbox', {axis:true, boundingbox:[-1/2,1/2,1/2,-1/2]}); | ||
+ | |||
+ | var g = board.create('functiongraph', ["2*sin(1/x) - cos(1/x)"], {strokeColor: 'red'}); | ||
+ | var f = board.create('functiongraph', ["x^2*sin(1/x)"], {strokeWidth:2}); | ||
+ | </source> | ||
+ | |||
+ | Second example: | ||
+ | <source lang="javascript"> | ||
+ | var board = JXG.JSXGraph.initBoard('jxgbox2', {axis:true, boundingbox:[-1/2,0.08,1.5,-0.02]}); | ||
+ | |||
+ | var g_der = board.create('functiongraph', ["(0 < x && x < 1) ? ((sin((1 / ((PI * x) * (1 - x)))) * ((2 * (x * ((1 - x)^2))) - (2 * ((x^2) * (1 - x))))) - (((x^2) * ((1 - x)^2)) * (cos((1 / ((PI * x) * (1 - x)))) * (((PI * (1 - x)) - (PI * x)) / (((PI * x) * (1 - x))^2))))) : 0"], {strokeColor: 'red'}); | ||
+ | var g = board.create('functiongraph', ["(0 < x && x < 1) ? x^2*(1-x)^2*sin(1/(PI* x*(1-x))) : 0"], {strokeWidth:2}); | ||
</source> | </source> | ||
[[Category:Examples]] | [[Category:Examples]] | ||
[[Category:Calculus]] | [[Category:Calculus]] |
Latest revision as of 23:01, 23 March 2021
Consider the function (blue curve)
- [math] f: \mathbb{R} \to \mathbb{R}, x \mapsto \begin{cases} x^2\sin(1/x),& x\neq 0,\\ 0,& x=0\,. \end{cases} [/math]
[math]f[/math] is a continous and differentiable function. The derivative of [math]f[/math] is the function (red curve)
- [math] f': \mathbb{R} \to \mathbb{R}, x \mapsto \begin{cases} 2x\sin(1/x) - \cos(1/x), &x \neq 0,\\ 0,& x=0\,. \end{cases} [/math]
We observe that [math]f'(0) = 0[/math], but [math]\lim_{x\to0}f'(x)[/math] does not exist.
Therefore, [math]f'[/math] is an example of a derivative which is not continuous.
Here is another example:
- [math] g: \mathbb{R} \to \mathbb{R}, x \mapsto \begin{cases} x^2(1-x)^2\sin(1/(\pi x(1-x)),& 0 \lt x \lt 1\\ 0,& \mbox{otherwise} \end{cases}\,. [/math]
The underlying JavaScript code
First example:
var board = JXG.JSXGraph.initBoard('jxgbox', {axis:true, boundingbox:[-1/2,1/2,1/2,-1/2]});
var g = board.create('functiongraph', ["2*sin(1/x) - cos(1/x)"], {strokeColor: 'red'});
var f = board.create('functiongraph', ["x^2*sin(1/x)"], {strokeWidth:2});
Second example:
var board = JXG.JSXGraph.initBoard('jxgbox2', {axis:true, boundingbox:[-1/2,0.08,1.5,-0.02]});
var g_der = board.create('functiongraph', ["(0 < x && x < 1) ? ((sin((1 / ((PI * x) * (1 - x)))) * ((2 * (x * ((1 - x)^2))) - (2 * ((x^2) * (1 - x))))) - (((x^2) * ((1 - x)^2)) * (cos((1 / ((PI * x) * (1 - x)))) * (((PI * (1 - x)) - (PI * x)) / (((PI * x) * (1 - x))^2))))) : 0"], {strokeColor: 'red'});
var g = board.create('functiongraph', ["(0 < x && x < 1) ? x^2*(1-x)^2*sin(1/(PI* x*(1-x))) : 0"], {strokeWidth:2});